We study the 2-parameter acoustic Born series for an actual medium
with constant velocity and a density distribution. Using a
homogeneous background we define a perturbation, the difference
between actual and reference medium (we use background and reference
as synonyms), which exhibits an anisotropic behavior due to the
density distribution. For an actual medium with a constant velocity,
the reference velocity can be selected so that the waves in the
actual medium travel with the same speed as the waves in the
background medium. Scattering theory decomposes the actual wave
field into an infinite series where each term contains the
perturbation and the propagators in the background medium. Hence, in
this formalism, all propagations occur in the background medium and
the actual medium is included only through the perturbations which
scatter the propagating waves. The density-only perturbation has an isotropic and an anisotropic
component. The anisotropic component is dependent on the incident
direction of the propagating waves and behaves as a
purposeful perturbation in the sense that it annihilates
the part of the Born series that acts to correct the time to build
the actual wave field, an unnecessary activity when the reference
velocity is equal to the one in the actual medium. This means that
the forward series is not attempting to correct for an issue that
does not exist. We define the purposeful perturbation concept as the
intrinsic knowledge of precisely what a given term is designed to
accomplish. This is a remarkable behavior for a formalism that
predicts the scattered wave field with an infinite series. At each
order of approximation the output of the series is consistent with
the fact that the time is correct because the velocity is always
constant. In the density-only perturbation, the forward series only
seeks to predict the correct amplitudes. Finally, we extend the analysis to a wave propagating in a medium
where both density and velocity change. By selecting a convenient
set of parameters, we find a conceptual framework for the
multiparameter Born series. This framework provides an insightful
analysis that can be mapped and applied to the concepts and
algorithms of the inverse scattering series.